Finite Group Databasebeta

← #32701

Subgroups — #32701

Order 256.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2104C212yesmaximal, complemented
2#2032C212yesmaximal, complemented
3#2030C211yesmaximal, characteristic, complemented
4#2022C211yesmaximal, characteristic
5#1913C212yesmaximal
6#1899C212yesmaximal, complemented
7#1890C212yesmaximal, complemented
8#1828C212yesmaximal, complemented
9#1820C211yesmaximal, characteristic, complemented
10#543C2211yescharacteristic
11#534C2211yescharacteristic
12C2×#123C2212yes—
13#518C2212yes—
14C2×#117C2211yescharacteristic
15C2×#99C2212yes—
16C2×#99C2212yes—
17C2×#99C2212yes—
18C2×#99C2212yes—
19#381C2211yescharacteristic
20#381C2211yescharacteristic
21#381C2212yes—
22#342C2212yes—
23#342C2212yes—
24#342C2211yescharacteristic
25#342C2212yes—
26#342C2212yes—
27#342C2211yescharacteristic
28#342C2212yes—
29#342C2212yes—
30#342C2212yes—
31C22×D4C2312yes—
32C23×C4C2311yescharacteristic
33#123—22no—
34#123—22no—
35#123—22no—
36#123—42no—
37C22⋊Q8—41no—
38C22⋊Q8—42no—
39#121—41no—
40#120—22no—
41C4×D4—42no—
42C4×D4—42no—
43#117—22no—
44#117—41no—
45#117—41no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4C2312yes—
51C2×C22⋊C4C2312yes—
52C2×C22⋊C4C2312yes—
53C2×C22⋊C4C2312yes—
54C2×C22⋊C4C2311yescharacteristic
55C2×C22⋊C4C2311yescharacteristic
56C2×C22⋊C4C2312yes—
57C2×C42—21no—
58#99—42no—
59#99—42no—
60#99—42no—
61#99—42no—
62#95—41no—
63#95—41no—
64#95—42no—
65C24#4112yes—
66C24C2411yescharacteristic
67C2×Q8—41no—
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—41no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—42no—
87C22×C4#4111yescharacteristic
88C22×C4#4112yes—
89C22×C4#4111yescharacteristic
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—43no—
96C22⋊C4—22no—
97C22⋊C4—22no—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—22no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—42no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C42—42no—
113C42—42no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23—81no—
118C23—82no—
119C23C2×#4112yes—
120C23C2×#4111yescharacteristic
121C23H(5,𝔽2)11yescharacteristic
122C23H(5,𝔽2)11yescharacteristic
123C23H(5,𝔽2)12yes—
124Q8—161no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—161no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—81no—
154C2×C4—162no—
155C22—41no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—42no—
160C22—42no—
161C22—82no—
162C22—82no—
163C22—82no—
164C22#53811yescharacteristic
165C22#53812yes—
166C22#55112yes—
167C22#55111yescharacteristic
168C22#56011yescharacteristic
169C4—41no—
170C4—41no—
171C4—42no—
172C4—82no—
173C4—82no—
174C4—82no—
175C4—82no—
176C4—81no—
177C4—82no—
178C4—81no—
179C2—21no—
180C2—22no—
181C2—21no—
182C2—21no—
183C2—22no—
184C2—82no—
185C2#276011yescharacteristic
subgroupG/H
center, socleC2#2760
commutator, Frattini, cosocleC24C24
prime radical#534C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #2104C2semidirect
2. #2032C2semidirect
3. #2030C2semidirect
6. #1899C2semidirect
7. #1890C2semidirect
8. #1828C2semidirect
9. #1820C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#21042
#20322
#20302
#20222
#19132
#18992
#18902
#18282
#18202